Riemann Xi function


In mathematics, the Riemann Xi function is a variant of the Riemann zeta function, and is defined so as to have a particularly simple functional equation. The function is named in honour of Bernhard Riemann.

Definition

Riemann's original lower-case "xi"-function, was renamed with an upper-case by Edmund Landau. Landau's lower-case is defined as
for. Here denotes the Riemann zeta function and is the Gamma function. The functional equation for Landau's is
Riemann's original function, rebaptised upper-case by Landau, satisfies
and obeys the functional equation
Both functions are entire and purely real for real arguments.

Values

The general form for positive even integers is
where Bn denotes the n-th Bernoulli number. For example:

Series representations

The function has the series expansion
where
where the sum extends over ρ, the non-trivial zeros of the zeta function, in order of.
This expansion plays a particularly important role in Li's criterion, which states that the Riemann hypothesis is equivalent to having λn > 0 for all positive n.

Hadamard product

A simple infinite product expansion is
where ρ ranges over the roots of ξ.
To ensure convergence in the expansion, the product should be taken over "matching pairs" of zeroes, i.e., the factors for a pair of zeroes of the form ρ and 1−ρ should be grouped together.

Further references